# Homework Help: Photochemistry and Dipole-Dipole interactions

1. Oct 5, 2013

### Verdict

Let me begin by saying that this question had a disclaimer, saying that part a should NOT be done exactly, but in a back of the napkin manner. The idea is to make reasonable assumptions and approximations in order to obtain an answer.

1. The problem statement, all variables and given/known data
Suppose that a certain chemical synthesis of molecular aggregates is driven by light (photochemistry). Infrared photon absorption changes a large molecule's arrangement from almost nonpolar into a polarized configuration. These newly polarized molecules are then attracted to one another to form aggregates ("supramolecular assembly"). In free space, their dipolar electric potentials have the usual relation to distance away from the dipole (polarized molecule). However, chemistry is rarely performed in free-space...
Assume instead that the synthesis is performed in a 6.0 milli-Molar aqueous solution of NaCl (salt water) at room temperature. While water's dielectric constant will alter the field, the ionic concentrations will also vary to effectively screen the dipolar field.

Part a. What is the distance dependence of the interaction energy for two dipoles (assumed to be aligned) in this aqueous environment? (Here, you may simplify things by also assuming the distance dependent terms from the dipole-dipole interaction and from the ionic charge screening can simply be multiplied together.

Part b. Assume that the dipole's magnitude can be approximated as half an electron's charge separated by 3.8 nm. Make a rough estimate of the distance at which two polarized molecules' attractive potential will be greater than thermal energy, kBT = 25 meV.

Part c. Assume these photo-active molecules have a Stoke's radius of 2.4 nm. Under continuous
IR exposure (you may assume that exactly half of the large molecules are excited to the excited polar configuration), what is the approximate concentration (molecules per unit volume) of these molecules such that it takes them on average 55 seconds to diuse far enough to fall within the bonding distance of another polarized molecule (which you calculated in Part b)?

2. Relevant equations
Relevant to this are the Stokes-Einstein equation for part C
$$D = \frac{k_{B}T}{4 \pi \eta a}$$
The interaction energy between two dipoles (in vacuum)

3. The attempt at a solution

Part a:
This is actually the part I have the most difficulties with. I've already found the expression for the distance dependence of two dipoles (as given in the equations section) in vacuum, so that goes like 1/r^3. However, I'm quite unsure about the cosine terms. The question says the dipoles are aligned, so I'd say that means that θ1=θ2 (or does it mean that the negative of the one faces the positive of the other?. The phi I do not know.
Furthermore, I know (from my book, Lindsay's introduction to nanoscience) that this energy is reduced by an amount equal to the inverse of the dielectric constant of the medium they are immersed in, so that of water. However, this is where I get lost. The question clearly indicates that I have to do something with ionic charge screening, but I am clueless as to what this entails.
What I have done already (and I am NOT sure if it is of any use) is the following:
The solution is 6 milli molar. This means that per liter, there are 0.06 moles of NaCl in the solution. 1 mole is 6.02214×10^23 particles, so 6 milli molar is 3.61×10^22 particles per liter and 3.61×10^25 molecules per cubic meter.

So from here on, I do not know what to do. Any hints would be very much appreciated.

Edit: Upon closer inspection of the book, I did find a section that talks about this screening. It is described by the Poisson Boltzman equation, which (in the taylor approximation) can be solved

This is a good thing, as it uses the molarity. I'm not sure how to exactly proceed here. I can work out what all the factors are in this, ending up with a new expression for the energy, but I still don't know how to handle the angles and such. And should I still divide by the dielectric moment of water, if I'm using this equation (which already takes it into account)?

Part B
What I've done here (lacking a solution for A as of now) is the following.
Assuming that the dipole’s magnitude can be approximated as half an electron charge separated by 3.8 nm, the magnetic dipole moment is equal to μ=qx = 0.5×e×3.8×10^-9 = 3.04381x10^-28 C m.

Then, I want to do something with the equation I obtain in A, and just set it equal to the 25 meV and solve for r. That much I can do of course, but without anything from A I get stuck here.

Part C
Using the Einstein Stokes relation I get that, given is that the synthesis is performed at room temperature, T = 293 K, the viscosity of water at room temperature is 1.002 ×10^(-3) Pa s and the stokes radius is 2.4 nm, that the diffusion constant D = 8.9242×10^(-11) m² s^(-1).

Now, beyond this I will need a value of r from question B. However, that does not finish the question, I have to do something with the concentration. This, I also do not know how to do as of yet. I half expect having to do something with an expectation value, as the question asks to do this on average, but I have yet to figure this out. However, I'd first and foremost like to solve A, so that is what my question is about primarily.